Libraries

Include

Algorithms

Data

Plotting

Modelling framework

Data

We don't need much data, but having something to put in our plots would be handy.

Experimental data

Simulation data

Motivation plot

This plot visualises the problem. Due to the specific dynamics in the niche wt has more output from stem cells early in life. Past 400 days of age however, due to the lack of increased quiescence, knockout mice have much more active stem cells and thus higher output.

Regular Kalamakis model

Definition

Previously estimated parameters

Solutions

Parameters

Differences from other notebook.

We take the intervention models that we previously built. Further, we assume that $b$ is a function $\text{NSC}\rightarrow b$ with two parameters, one of which is dependent on genotype. More specifically, $$ b(\text{NSC}) = \frac{1}{1+\left(\frac{K_{ab}}{\text{NSC}}\right)^{n_b}} $$ Where $K_{ab}$ is the shared half-probability population around 150 stem cells and $n_b$ the genotype dependent shape parameter.

A place to store metrics

Beta/self model

Takes parameters, outputs simulations. Plots simulations with datapoints. Thus needs:

In this model the effect of interferon is a changed $r'$, as well as a changedl $b_n$. This is assuming that in wt: $$ r'_\text{wt}(t) \neq 0, $$ i.e. $ \beta_r \neq 0. $ and in ko $$ r'_\text{ko}(t) = 0, $$ i.e. $ \beta_r = 0. $

Since $r$ is exponential this means that this model should for some intervention timepoint $t_1$ have a $r$ of the form: $$ \frac{\text{d}}{\text{d}t}r(t) = \begin{cases} -\beta_r r(t)& \text{if }t<t_1 \\ 0 & \text{else} \end{cases} $$

Thus we have parameters: $$ \theta = ( \text{NSC}_0, r_0, \beta_r, K_{ab}, n_{b,wt}, n_{b,ko}, p_s, t_1 ) $$

Self-renewal depends on genotype

Self-renewal is partially genotype dependent

Self-renewal is genotype invariant

Value/self model

In this model the effect of interferon is a changed $r$.

This just means that we need a two-stage $r(t)$ with parameters $\theta_r = (r_0, \beta_r, r_1, t_1)$ which then is $$ r(t) = \begin{cases} r_0 \cdot e^{-\beta_rt} & \text{if } t < t_1\\ r_1 & \text{else}\\ \end{cases} $$

Self-renewal depends on genotype

Self-renewal is partially genotype dependent

Self-renewal is genotype-invariant

Big comparison

Blub